Torsion in the full orbifold K-theory of abelian symplectic quotients

TL;DR

This paper proves the absence of additive torsion in the integral full orbifold K-theory of certain symplectic quotients, extending Morse-Bott techniques and applying to toric orbifolds and GKM spaces.

Contribution

It introduces conditions ensuring torsion-free orbifold K-theory in symplectic quotients, extending Morse-Bott analysis to moment map level sets.

Findings

Integral full orbifold K-theory is torsion-free for many symplectic quotients.

Extension of Morse-Bott functions to moment map components.

Application to low-rank coadjoint orbits of types A and B.

Abstract

Let (M,\omega,\Phi) be a Hamiltonian T-space and let H be a closed Lie subtorus of T. Under some technical hypotheses on the moment map \Phi, we prove that there is no additive torsion in the integral full orbifold K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S^1-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our results using low-rank coadjoint orbits of type A and B.